34.2.8 problem 8

Internal problem ID [6038]
Book : A treatise on ordinary and partial differential equations by William Woolsey Johnson. 1913
Section : Chapter 2, Equations of the first order and degree. page 20
Problem number : 8
Date solved : Wednesday, March 05, 2025 at 12:10:22 AM
CAS classification : [_separable]

\begin{align*} \sin \left (x \right ) \cos \left (y\right )&=\cos \left (x \right ) \sin \left (y\right ) y^{\prime } \end{align*}

✓ Maple. Time used: 0.028 (sec). Leaf size: 11

ode:=sin(x)*cos(y(x)) = cos(x)*sin(y(x))*diff(y(x),x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \arccos \left (\frac {\cos \left (x \right )}{c_1}\right ) \]

✓ Mathematica. Time used: 5.273 (sec). Leaf size: 47

ode=Sin[x]*Cos[y[x]]==Cos[x]*Sin[y[x]]*D[y[x],x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\arccos \left (\frac {1}{2} c_1 \cos (x)\right ) \\ y(x)\to \arccos \left (\frac {1}{2} c_1 \cos (x)\right ) \\ y(x)\to -\frac {\pi }{2} \\ y(x)\to \frac {\pi }{2} \\ \end{align*}

✓ Sympy. Time used: 0.503 (sec). Leaf size: 20

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(sin(x)*cos(y(x)) - sin(y(x))*cos(x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = - \operatorname {acos}{\left (C_{1} \cos {\left (x \right )} \right )} + 2 \pi , \ y{\left (x \right )} = \operatorname {acos}{\left (C_{1} \cos {\left (x \right )} \right )}\right ] \]